Abstract and Applied Analysis

Volume 2013 (2013), Article ID 154912, 4 pages

http://dx.doi.org/10.1155/2013/154912

## Sufficient Conditions for Non-Bazilevič Functions

^{1}School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455002, China^{2}School of Business, Central South University, Changsha, Hunan 410083, China

Received 11 July 2013; Accepted 19 September 2013

Academic Editor: Alberto Fiorenza

Copyright © 2013 Qun-Fa Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The main purpose of this paper is to derive some sufficient conditions for analytic functions to be of non-Bazilevič type.

#### 1. Introduction

Let denote the class of functions of the form
which are * analytic* in the * open* unit disk:

For and , a function is said to be in the class if it satisfies the condition As usual, the class is said to be non-Bazilevič functions of order (see [1]).

For some recent investigations of non-Bazilevič functions, see, for example the works of [2–6] and the references cited therein.

For two functions and , analytic in , we say that the function is subordinate to in and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence:

To derive our main results, we need the following lemmas.

Lemma 1 (see [7]). * Let be analytic in and let be analytic and starlike (with respect to the origin) univalent in with . If
**
then
*

Lemma 2 (see [8]). * Let be univalent in . Also let be analytic in the domain containing with when . Set
**
Suppose that *(1)* is starlike univalent in ;*(2)* for . **If is analytic in with , and
**
then , and is the best dominant.*

Lemma 3 (see [9]). * Let be a set in the complex plane and suppose that is a mapping from to which satisfies for and for all real such that .**If the function is analytic in and for all , then .*

In this paper, we aim at proving some sufficient conditions for analytic functions to be of non-Bazilevič type.

#### 2. Main Results

Our first main result is given by Theorem 4.

Theorem 4. *Suppose that is starlike in with . If
**
then
*

*Proof. *We define the function by
Then is analytic in with . It follows from (15) that
Combining (13) and (16), we find that
By Lemma 1, we deduce that
From (15) and (18), we readily get the assertion (14) of Theorem 4.

Theorem 5. *If satisfies the inequality**
then .*

*Proof. *Suppose that the function is defined by (15). It follows that
Combining (19) and (20), we know that
An application of Lemma 1 to (21) yields
By noting that
which implies that the region is symmetric with respect to the real axis and is convex univalent in therefore, we have

Combining (15), (22), and (24), we conclude that
This completes the proof of Theorem 5.

Theorem 6. *Suppose that is convex in with . If
**
then
**
and is the best dominant.*

*Proof. *Suppose that the function is defined by (15). It follows that

We now assume that
Obviously, and are analytic in the plane. By noting that the function
is starlike in and
it follows from (26) that
Combining (27), (29), and Lemma 2, we get the assertion of Theorem 6.

*Remark 7. *By taking suitable and in Theorems 4 and 6, respectively, we can get some useful consequences. Here we choose to omit the details.

Theorem 8. *If satisfies the condition**
then .*

*Proof. *Suppose that
Then is analytic in . It follows from (35) that
where
For all real and satisfying , we have

We now put
Then for all real such that . Moreover, in view of (34), we know that . Thus, by Lemma 3, we deduce that
which shows that the desired assertion of Theorem 8 holds.

#### Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008, 11226088, 71171024, 71371195, and 70921001 and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant no. 12A110002 of the People’s Republic of China.

#### References

- M. Obradović, “A class of univalent functions,”
*Hokkaido Mathematical Journal*, vol. 27, no. 2, pp. 329–335, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Obradović, S. Ponnusamy, and P. Vasundhra, “Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions,”
*Indian Academy of Sciences. Proceedings Mathematical Sciences*, vol. 119, no. 5, pp. 593–610, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. N. Shanmugam, S. Sivasubramanian, M. Darus, and S. Kavitha, “On sandwich theorems for certain subclasses of non-Bazilevič functions involving Cho-Kim transformation,”
*Complex Variables and Elliptic Equations*, vol. 52, no. 10-11, pp. 1017–1028, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. N. Shanmugam, S. Sivasubramanian, B. A. Frasin, and S. Kavitha, “On sandwich theorems for certain subclasses of analytic functions involving Carlson-Shaffer operator,”
*Journal of the Korean Mathematical Society*, vol. 45, no. 3, pp. 611–620, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Tuneski and M. Darus, “Fekete-Szegö functional for non-Bazilevič functions,”
*Acta Mathematica Academiae Paedagogicae Nyíregyháziensis (New Series)*, vol. 18, no. 2, pp. 63–65, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wang, C. Gao, and M. Liao, “On certain generalized class of non-Bazilevič functions,”
*Acta Mathematica Academiae Paedagogicae Nyíregyháziensis (New Series)*, vol. 21, no. 2, pp. 147–154, 2005. View at Google Scholar · View at MathSciNet - D. Yang, “Some criteria for multivalently starlikeness,”
*Southeast Asian Bulletin of Mathematics*, vol. 24, no. 3, pp. 491–497, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. S. Miller and P. T. Mocanu,
*Differential Subordinations, Theory and Applications*, vol. 225, Marcel Dekker, New York, NY, USA, 2000. View at MathSciNet - S. S. Miller and P. T. Mocanu, “Differential subordinations and inequalities in the complex plane,”
*Journal of Differential Equations*, vol. 67, no. 2, pp. 199–211, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet